Generates the convex regression spline (called C-spline) basis matrix by integrating I-spline basis for a polynomial spline or the corresponding derivatives.

## Usage

cSpline(
x,
df = NULL,
knots = NULL,
degree = 3L,
intercept = TRUE,
Boundary.knots = NULL,
derivs = 0L,
scale = TRUE,
...
)

## Arguments

x

The predictor variable. Missing values are allowed and will be returned as they are.

df

Degree of freedom that equals to the column number of the returned matrix. One can specify df rather than knots, then the function chooses df - degree - as.integer(intercept) internal knots at suitable quantiles of x ignoring missing values and those x outside of the boundary. If internal knots are specified via knots, the specified df will be ignored.

knots

The internal breakpoints that define the splines. The default is NULL, which results in a basis for ordinary polynomial regression. Typical values are the mean or median for one knot, quantiles for more knots.

degree

The degree of C-spline defined to be the degree of the associated M-spline instead of actual polynomial degree. For example, C-spline basis of degree 2 is defined as the scaled double integral of associated M-spline basis of degree 2.

intercept

If TRUE by default, all of the spline basis functions are returned. Notice that when using C-Spline for shape-restricted regression, intercept = TRUE should be set even when an intercept term is considered additional to the spline basis in the model.

Boundary.knots

Boundary points at which to anchor the splines. By default, they are the range of x excluding NA. If both knots and Boundary.knots are supplied, the basis parameters do not depend on x. Data can extend beyond Boundary.knots.

derivs

A nonnegative integer specifying the order of derivatives of C-splines. The default value is 0L for C-spline basis functions.

scale

A logical value indicating if scaling C-splines is required. If TRUE by default, each C-spline basis is scaled to have unit height at right boundary knot. The corresponding I-spline and M-spline produced by deriv methods will be scaled to the same extent.

...

Optional arguments that are not used.

## Value

A numeric matrix of length(x) rows and df columns if

df is specified or length(knots) + degree + as.integer(intercept) columns if knots are specified instead. Attributes that correspond to the arguments specified are returned mainly for other functions in this package.

## Details

It is an implementation of the closed-form C-spline basis derived from the recursion formula of I-splines and M-splines.

## References

Meyer, M. C. (2008). Inference using shape-restricted regression splines. The Annals of Applied Statistics, 2(3), 1013--1033.

iSpline for I-splines; mSpline for M-splines.

## Examples

library(splines2)

x <- seq.int(0, 1, 0.01)
knots <- c(0.3, 0.5, 0.6)

### when 'scale = TRUE' (by default)
csMat <- cSpline(x, knots = knots, degree = 2)

op <- par(mar = c(2.5, 2.5, 0.2, 0.1), mgp = c(1.5, 0.5, 0))
matplot(x, csMat, type = "l", ylab = "C-spline basis")
abline(v = knots, lty = 2, col = "gray")

isMat <- deriv(csMat)
msMat <- deriv(csMat, derivs = 2)
matplot(x, isMat, type = "l", ylab = "scaled I-spline basis")

matplot(x, msMat, type = "l", ylab = "scaled M-spline basis")

## reset to previous plotting settings
par(op)

### when 'scale = FALSE'
csMat <- cSpline(x, knots = knots, degree = 2, scale = FALSE)

## the corresponding I-splines and M-splines (with same arguments)
isMat <- iSpline(x, knots = knots, degree = 2)
msMat <- mSpline(x, knots = knots, degree = 2, intercept = TRUE)

## or using deriv methods (more efficient)
isMat1 <- deriv(csMat)
msMat1 <- deriv(csMat, derivs = 2)

## equivalent
stopifnot(all.equal(isMat, isMat1, check.attributes = FALSE))
stopifnot(all.equal(msMat, msMat1, check.attributes = FALSE))